On the autoepistemic reconstruction of logic programming

In his paper [12] Ross extends the well-founded semantics for normal logic programs [16] to disjunctive logic programs. His definition is top-down and it is closer to a procedural semantics than to the elegant fixpoint definition of the well-founded semantics for normal logic programs. In the present paper, we propose a declarative, bottom-up fixpoint definition of the well-founded semantics for disjunctive logic programs. Our construction of the greatest unfounded set of extended literals is similar with the construction of the greatest unfounded set for normal programs. As a consequence, the connection between the well-founded semantics for normal programs and the well-founded semantics for disjunctive programs is made clearer.KeywordsLogic ProgramDeductive DatabaseNormal ProgramDisjunctive ProgramExtended AtomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Over recent years, various semantics have been proposed for dealing with updates in the setting of logic programs. The availability of different semantics naturally raises the question of which are most adequate to model updates. A systematic approach to face this question is to identify general principles against which such semantics could be evaluated. In this paper we motivate and introduce a new such principle – the refined extension principle – which is complied with by the stable model semantics for (single) logic programs. It turns out that none of the existing semantics for logic program updates, even though based on stable models, complies with this principle. For this reason, we define a refinement of the dynamic stable model semantics for Dynamic Logic Programs that complies with the principle.

M. Bezem defined an extensional semantics for positive higher-order logic programs. Recently, it was demonstrated by Rondogiannis and Symeonidou that Bezem's technique can be extended to higher-order logic programs with negation, retaining its extensional properties, provided that it is interpreted under a logic with an infinite number of truth values. Rondogiannis and Symeonidou also demonstrated that Bezem's technique, when extended under the stable model semantics, does not in general lead to extensional stable models. In this paper, we consider the problem of extending Bezem's technique under the well-founded semantics. We demonstrate that the well-founded extensionfailsto retain extensionality in the general case. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting cannot distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.

Three-valued completion for abductive logic programs

Most of the theoretical work on the semantics of logic programs assumes an interpreter that provides a complete resolution procedure. In contrast, for reasons of efficiency, most logic programming languages are built around incomplete procedures. This difference is rooted in Prolog, which evaluates resolvent trees in a depth-first rather than a breadth-first order. The gap is widened by some equational logic languages, which combine the incompleteness of depth-first evaluation with incomplete approximations to equational unification. Because of this gap, it is unsound to reason about logic programs using their declarative semantics. This in turn makes it difficult to develop abstraction mechanisms that can be used to partition a logic program into independently specifiable modules.

This paper extends fuzzy logic programs [12, 24] to allow the explicit representation of classical negation as well as non-monotonic negation, by introducing the notion of extended fuzzy logic programs. We present the fuzzy answer set semantics for the extended fuzzy logic programs, which is based on the classical answer set semantics of classical extended logic programs [7]. We show that the proposed semantics is a natural extension to the classical answer set semantics of classical extended logic programs [7]. Furthermore, we define fixpoint semantics for extended fuzzy logic programs with and without non-monotonic negation, and study their relationship to the fuzzy answer set semantics. In addition, we show that the fuzzy answer set semantics is reduced to the stable fuzzy model semantics for normal fuzzy logic programs introduced in [42]. The importance of that is computational methods developed for normal fuzzy logic programs can be applied to the extended fuzzy logic programs. Moreover, we show that extended fuzzy logic programs can be intuitively used for representing and reasoning about actions in fuzzy environment.

Part of the theory of logic programming and nonmonotonic reasoning concerns the study of fixed-point semantics for these paradigms. Several different semantics have been proposed during the last two decades, and some have been more successful and acknowledged than others. The rationales behind those various semantics have been manifold, depending on one's point of view, which may be that of a programmer or inspired by commonsense reasoning, and consequently the constructions which lead to these semantics are technically very diverse, and the exact relationships between them have not yet been fully understood. In this paper, we present a conceptually new method, based on level mappings, which allows to provide uniform characterizations of different semantics for logic programs. We will display our approach by giving new and uniform characterizations of some of the major semantics, more particular of the least model semantics for definite programs, of the Fitting semantics, and of the well-founded semantics. A novel characterization of the weakly perfect model semantics will also be provided.

Although negative conclusions are presented implicitly in Normal Hybrid Probabilistic Programs (NHPP) [26] through the closed world assumption, representing and reasoning with explicit negation is needed in NHPP to allow the ability to reason with incomplete knowledge. In this paper we extend the language of NHPP to explicitly encode classical negation in addition to non-monotonic negation. The semantics of the extended language is based on the answer set semantics of traditional logic programming [9]. We show that the proposed semantics is a natural extension to the answer set semantics of traditional logic programming [9]. In addition, the proposed semantics is reduced to stable probabilistic model semantics of NHPP [26]. The importance of that is computational methods developed for NHPP can be applied to the proposed language. Furthermore, we show that some commonsense probabilistic knowledge can be easily represented in the proposed language.KeywordsLogic ProgramLogic ProgrammingClassical NegationIncomplete KnowledgeProbability IntervalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given a first order language L, and a notion of a logic L w.r.t. L, we investigate the topological properties of the space of L-structures for L. We show that under a topology called the query topology which arises naturally in logic programming, the space of L-models (where L is a decent logic) of any sentence (set of clauses) in L may be regarded as a (closed, compact) T4-space. We then investigate the properties of maps from structures to structures. Our results allow us to apply various well-known results on the fixed-points of operators on topological spaces to the semantics of logic programming – in particular, we are able to derive necessary and sufficient topological conditions for the completion of covered general logic programs to be consistent. Moreover, we derive sufficient conditions guaranteeing the consistency of program completions, and for logic programs to be determinate. We also apply our results to characterize consistency of the unions of program completions.

We present some results on semantics for logic programs. We interpret logic programs in Kripke models. They connect the area of logic programming and that of nonclassical logics. New connectives are carried over dynamic logic to logic programs, resulting in so-called predicate programs. A wide class of predicate programs is shown to have the same meaning as formulas of dynamic logic with context-free programs. This allows to discuss the known problem of introducing the negation into logic programs in new light.

Logic programming has developed as a rich field, built over a logical substratum whose main constituent is a nonclassical form of negation, sometimes coexisting with classical negation. The field has seen the advent of a number of alternative semantics, with Kripke–Kleene semantics, the well-founded semantics, the stable model semantics, and the answer-set semantics standing out as the most successful. We show that all aforementioned semantics are particular cases of a generic semantics, in a framework where classical negation is the unique form of negation and where the literals in the bodies of the rules can be ‘marked’ to indicate that they can be the targets of hypotheses. A particular semantics then amounts to choosing a particular marking scheme and choosing a particular set of hypotheses. When a literal belongs to the chosen set of hypotheses, all marked occurrences of that literal in the body of a rule are assumed to be true, whereas the occurrences of that literal that have not been marked in the body of the rule are to be derived in order to contribute to the firing of the rule. Hence the notion of hypothetical reasoning that is presented in this framework is not based on making global assumptions, but more subtly on making local, contextual assumptions, taking effect as indicated by the chosen marking scheme on the basis of the chosen set of hypotheses. Our approach offers a unified view on the various semantics proposed in logic programming, classical in that only classical negation is used, and links the semantics of logic programs to mechanisms that endow rule-based systems with the power to harness hypothetical reasoning.

The Herbrand model H of a definite logic program P is an initial model among the class of all the models of P, interpreting P as an initial theory. Such a theory (program) proves (computes) only positive literals (atoms) in P, so it does not deal with negation. In this paper, we introduce isoinitial semantics for logic programs and show that it can provide a rich semantics for logic programs, which can deal with not just negation, but also incomplete information, parametricity and compositionality.KeywordsLogic ProgramLogic ProgrammingOpen ProgramConstant SymbolGround AtomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce a new semantics for logic programming languages. It generalises the traditional Herbrand universe semantics, and specialises the semantics of logical relations, as used in analysing parametricity in functional and imperative programming languages. We outline a typed logic programming language, give it this semantics, and show how it supports structured development of logic programs as advocated by Sterling et al. In particular, it gives semantics for some dynamic aspects of logic programs.

It is well known that, under certain conditions, it is possible to split logic programs under stable model semantics, i.e. to divide such a program into a number of different levels, such that the models of the entire program can be constructed by incrementally constructing models for each level. Similar results exist for other non-monotonic formalisms, such as auto-epistemic logic and default logic. In this work, we present a general, algebraic splitting theory for programs/theories under a fixpoint semantics. Together with the framework of approximation theory, a general fixpoint theory for arbitrary operators, this gives us a uniform and powerful way of deriving splitting results for each logic with a fixpoint semantics. We demonstrate the usefulness of these results, by generalizing Lifschitz and Turner's splitting theorem to other semantics for (non-disjunctive) logic programs.

Two natural ways to specify the declarative semantics of logic programs and deductive databases are the fixpoint theory of Van Emden and Kowalski [11] and Clark's predicate completion [2]. The fixpoint theory does not apply to general programs with negation; a generalization of the theory can be defined [1,12] only if the programs are stratified. Clark's predicate completion is defined for logic programs with negation. In general, it fails to capture their intended semantics [7,8,9].In this paper, we introduce a new notion of quasi-interpretation as a set of ground clauses of the form A ← ¬ B 1, ..., ← B n and extend the classic fixed point theory in [11] to quasi-interpretations. The semantics of a logic program P is defined by Clark's predicate completion of the least fixpoint of a continuous operator T P on quasi-interpretations. It is called the fixpoint completion of P, fixcomp(P). We then discuss the relations between fixcomp(P) and other approaches [5,7,8,9].KeywordsLogic ProgramStable ModelModel SemanticPerfect ModelDeductive DatabaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.